A090184 Number of partitions of the n-th 3-smooth number into parts 2 and 3.

0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 5, 6, 7, 9, 10, 11, 13, 14, 17, 19, 22, 25, 28, 33, 37, 41, 43, 49, 55, 65, 73, 82, 86, 97, 109, 122, 129, 145, 163, 171, 193, 217, 244, 257, 289, 325, 342, 365, 385, 433, 487, 513, 577, 649, 683, 730, 769, 865, 973, 1025, 1094, 1153

Offset: 1

Author

Reinhard Zumkeller, Jan 21 2004

Keywords

• nonn

Examples

			n=11: A003586(11) = 2^3 * 3 = 24: 3+3+3+3+3+3+3+3 = 3+3+3+3+3+3+2+2+2 = 3+3+3+3+2+2+2+2+2+2 = 3+3+2+2+2+2+2+2+2+2+2 = 2+2+2+2+2+2+2+2+2+2+2+2: a(11)=5.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A022328, A022329.

Cf. A003586, A008615.

Cf. A103221, A117222, A117220, A117221.

Programs

Mathematica

smooth3Q[n_] := n/2^IntegerExponent[n, 2]/3^IntegerExponent[n, 3] == 1;
Length[IntegerPartitions[#, All, {2, 3}]]& /@ Select[Range[10000], smooth3Q] (* Jean-François Alcover, Oct 13 2021 *)
With[{nn = 6^5}, Map[Floor[#/2] - Floor[#/3] &, Union@ Flatten@ Table[2^a * 3^b, {a, 0, Log2[#]}, {b, 0, Log[3, #/(2^a)]}] &[nn] + 2]] (* Michael De Vlieger, Oct 13 2021 *)

Python

from sympy import integer_log
def A090184(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
    return ((m:=bisection(f,n,n)+2)>>1)-m//3 # Chai Wah Wu, Oct 22 2024

Formula

a(2^i * 3^j) = floor(2^(i-1) * 3^(j-1) + 1), i*j>0.

a(n) = A103221(A003586(n)).

Extensions

Offset changed to 1 by Alois P. Heinz, Oct 15 2021